Advances in Fuzzy Systems

Volume 2011 (2011), Article ID 768028, 5 pages

http://dx.doi.org/10.1155/2011/768028

## On Fuzzy *Sp*-Open Sets

^{1}Department of Mathematics, Rada'a College of Education and Science, Albida, Yemen^{2}Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

Received 29 December 2010; Accepted 18 February 2011

Academic Editor: Ibrahim Ozkan

Copyright © 2011 Hakeem A. Othman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new class of generalized fuzzy open sets in fuzzy topological space,
called fuzzy *sp*-open sets, are introduced, and their properties are studied and the relationship between this new concept and other weaker forms of fuzzy open sets we discussed. Moreover, we introduce the fuzzy *sp*-continuous (resp., fuzzy *sp*-open) mapping and other stronger forms of *sp*-continuous (resp., fuzzy *sp*-open) mapping and establish their various characteristic properties. Finally, we study the relationships between all these mappings and other weaker forms of fuzzy continuous mapping and introduce fuzzy *sp*-connected. Counter examples are given to show the noncoincidence of these sets and mappings.

#### 1. Introduction

In 1996, Dontchev and Przemski, [1] have introduced the concept of *sp*-open sets in general topology. In this paper, we extend the notion of *sp*-open sets to fuzzy topology space and study some notions based on this new concept. We further study the relation between fuzzy *sp*-open sets and other types of fuzzy open sets. We also introduce the concepts of fuzzy *sp*-continuous (resp., fuzzy *sp*-open) mapping, other stronger forms of fuzzy *sp*-continuous (resp., fuzzy *sp*-open) mapping, and discuss their relation with other weaker forms of fuzzy continuous mapping.

#### 2. Preliminaries

Throughout this paper, by or simply by we mean a fuzzy topological space (fts, shorty) and means a mapping from a fuzzy topological space to a fuzzy topological space . If is a fuzzy set and is a fuzzy singleton in , then , , , denote, respectively, the neighborhood system of , the interior of , the closure of , and complement of .

Now, we recall some of the basic definitions and results in fuzzy topology.

*Definition 2.1 (see [2]). *A fuzzy singleton in is a fuzzy set defined by: , for and otherwise, where . The point is said to have support and value .

*Definition 2.2. *A fuzzy set in a X is called fuzzy open [3] (resp., Fuzzy preopen [4], Fuzzy -open [5]) set if Int cl Int (resp., Int cl , cl Int cl , cl Int ). The family of all fuzzy -open (resp., fuzzy preopen, fuzzy -open, fuzzy semiopen) sets of is denoted by (resp., , , ).

*Definition 2.3 (see [4]). *Let be any fuzzy set. Then,(i)is called preclosure,(ii) is called pre-Interior.The definitions of , , , and are similar.

Theorem 2.4. *For any fuzzy set in a , the following statements are true:*(i)* and [6],*(ii)* [7].*

Theorem 2.5 (see [3, 4]). *The arbitrary union of fuzzy preopen (resp., fuzzy semiopen) sets is a fuzzy preopen (resp., fuzzy semiopen) set.*

*Definition 2.6. *A mapping is said to be(i)fuzzy -continuous [3] if is fuzzy -open set in for each fuzzy open set in ,(ii)fuzzy semi continuous [3] if is fuzzy semiopen set in for each fuzzy open set in ,(iii)fuzzy -continuous [5] if is fuzzy -open set in for each fuzzy open set in .

#### 3. Fuzzy *SP*-Open Set

*Definition 3.1. *A fuzzy subset of fuzzy space is called fuzzy *sp*-open set if . The class of all fuzzy *sp*-open sets in will be denoted be .

It is obvious that .

Proposition 3.2. *Let be fuzzy sp -open set such that . Then, is fuzzy preopen.*

Using Theorem 2.5, we can easily prove the next corollary.

Corollary 3.3. *Any union of fuzzy sp -open sets is a fuzzy sp-open set.*

*Remark 3.4. *The intersection of fuzzy *sp*-open sets need not be fuzzy *sp*-open set. This is illustrated by the following example.

*Example 3.5. *Let and , and be fuzzy sets of defined as
Let . Clearly, is a fuzzy topology on , and by easy computation, it follows that and are fuzzy *sp*-open sets. But is not a fuzzy fuzzy *sp*-open set.

Theorem 3.6. *For any fuzzy subset of a fuzzy space X, the following properties are equivalent:*(i)* is fuzzy sp-open,*(ii)*.*

*Proof. *. Let be a fuzzy *sp*-open, that is, . Then, we have

*Definition 3.7. *A fuzzy subset of fuzzy space is called fuzzy *sp*-closed set if . The class of all fuzzy *sp*-closed sets in will be denoted be .

*Definition 3.8. *Let any fuzzy set. Then,(i)is a fuzzy is called fuzzy *sp*-closure,(ii) is called fuzzy *sp*-Interior.By using Definitions 3.1, 3.7, and 3.8, we can prove the following theorems.

Theorem 3.9. *Let and be the fuzzy sets in . Then, the following statements hold*(i)* is fuzzy sp-closed,*(ii)*,*(iii)*,*(iv)* is fuzzy sp-open,*(v)*,*(vi)*,*(vii)*.*

Theorem 3.10. *For a fuzzy subset of a fuzzy space , the following statements are holding:*(i)*,*(ii)*,*(iii)*,*(iv)*.*

Theorem 3.11. *For any fuzzy subset of a fuzzy space X, the following statements are equivalent:*(i)* is fuzzy sp-closed, *(ii)* is fuzzy sp-open,*(iii)*, and *(iv)*.*

Theorem 3.12. *A fuzzy set in a fuzzy topology space is fuzzy sp -open if and only if for every fuzzy point , there exists a fuzzy sp -open set such that .*

*Proof. *If is a fuzzy *sp*-open set, then we may take for every .

Conversely, we have and, hence, . This shows that is a fuzzy *sp*-open set.

From Definitions 2.2 and 3.1, the above “Implication Figure 1” illustrates the relation of different classes of fuzzy open sets.

*Remark 3.13. *The converse of these relations need not be true, in general as shown by the following examples.

*Example 3.14. *Let and , and be fuzzy sets of defined as
Let . Clearly, is a fuzzy topological space on , and by easy computation, we can see:(i) is fuzzy *sp*-open set which is neither fuzzy -open set nor fuzzy preopen,(ii) is fuzzy *sp*-open which is not semiopen,(iii) is fuzzy *sp*-open set which is not fuzzy open.

*Example 3.15. *Let and fuzzy sets of defined as
Let . Clearly, is a fuzzy topological space on , and by easy computation, it follows that is fuzzy -open set which is not fuzzy *sp*-open.

#### 4. Fuzzy *SP*-Continuous Mapping

*Definition 4.1. *A mapping is said to be(i)fuzzy *sp*-continuous if is fuzzy *sp*-open set in for each fuzzy open set in ,(ii)fuzzy *sp*^{⋆}-continuous if is fuzzy *sp*-open set in for each fuzzy *sp*-open set in ,(iii)fuzzy *sp*^{⋆⋆}-continuous if is fuzzy open set in for each fuzzy *sp*-open set in .

Theorem 4.2. *For a mapping , the following statements are equivalent: *(i)* is fuzzy -continuous;*(ii)*for every fuzzy singleton in and every open set in such that , there exists a fuzzy sp -open set such that and ;*(iii)*for every fuzzy singleton in and every open set in such that , there exists a fuzzy sp -open set such that and ;*(iv)*the inverse image of each fuzzy closed set in is fuzzy sp-closed;*(v)* for each ;*(vi)* for each .*

*Proof. * (i)(ii) Let fuzzy singleton be in and every open set in such that , there exists a fuzzy open set be in such that . Since is *sp*-continuous, is fuzzy *sp*-open and we have or .

(ii)(iii) Let fuzzy singleton be in and every fuzzy open set be in such that , there exists a fuzzy *sp*-open such that and . So, we have and .

(iii)(i) Let be a fuzzy open set in and let us take . This shows that . Since is a fuzzy open set, then there exists a fuzzy *sp*-open set such that and . This shows that . By Theorem 3.12, it follows that is fuzzy *sp*-open set in and hence is fuzzy *sp*-continuous.

(i)(iv) Let be a fuzzy closed in . This implies that is fuzzy open set. Hence, is fuzzy *sp*-open set in , that is, is fuzzy *sp*-open set in . Thus, is a fuzzy *sp*-closed set in .

(iv)(v) Let , then is *sp*-closed in , that is, .

(v)(vi) Let , put in , then so that . This gives .

(vi)(i) Let , be fuzzy open set. put and , then , that is, -closed set in , so is *sp*-continuous mapping.

*Definition 4.3. *A mapping is said to be(i)fuzzy *sp*-open (Fuzzy *sp*-closed) if is fuzzy *sp*-open (fuzzy *sp*-closed) set in for each fuzzy open (fuzzy closed) set in ,(ii)fuzzy *sp*^{⋆}-open (Fuzzy *sp*^{⋆}-closed) if is fuzzy *sp*-open (fuzzy *sp*-closed) set in for each fuzzy -open (fuzzy *sp*-closed) set in ,(iii)fuzzy *sp*^{⋆⋆}-open (Fuzzy *sp*^{⋆⋆}-closed) if is fuzzy open (fuzzy closed) set in for each fuzzy *sp*-open (fuzzy -closed) set in .

*Remark 4.4. *If is fuzzy *sp*-continuous mapping and is fuzzy *sp*-continuous mapping, then may not be a fuzzy *sp*-continuous mapping; this can be show by the following example.

*Example 4.5. *Let and and be fuzzy sets of defined as,
Consider, , , and where , , and and the mapping and defined as and . It is clear that and are fuzzy *sp*-continuous mapping. But is not a fuzzy *sp*-continuous mapping. This because and is not fuzzy *sp*-open set, and hence is not fuzzy *sp*-continuous mapping.

Theorem 4.6. *If is fuzzy sp -continuous mapping and is fuzzy continuous mapping, then is fuzzy sp-continuous mapping.*

*Proof. *Let be a fuzzy set of . Then, . And because is fuzzy continuous this implies that is a fuzzy open set of and hence is a fuzzy *sp*-open set in . Therefore, is a fuzzy *sp*- continuous mapping.

From Definitions 4.1 and 4.3, we can have the above “Implication Figure 2” illustrates the relation between different classes of fuzzy *sp*-continuous (fuzzy semi *sp*-open) mappings.

The above “Implication Figure 3” illustrates the relation between fuzzy *sp*-continuous and different classes of fuzzy continuous mapping.

*Remark 4.7. *We can see the converse of these relations need not be true, in general as shown by the following examples.

*Example 4.8. *Let and , and fuzzy sets of defined as
Consider , , and where , , and and the mapping and defined as and . It is clear that:(i) is a fuzzy *sp*-continuous mapping which is neither fuzzy -continuous mapping nor fuzzy semi continuous mapping,(ii) is a fuzzy -continuous mapping which is not fuzzy *sp*-continuous mapping,(iii) is a fuzzy *sp*-continuous mapping which is neither fuzzy *sp*^{⋆}-continuous mapping nor fuzzy *sp*^{⋆⋆}-continuous mapping, and this is because is fuzzy *sp*-open in which is neither fuzzy *sp*-open nor fuzzy open in ,(iv)if defined as: , it is clear that is a fuzzy *sp*-open mapping which is neither fuzzy *sp*^{⋆}-open mapping nor fuzzy *sp*^{⋆⋆}-open mapping.

*Definition 4.9. *A fuzzy set in an is said to be fuzzy connected if cannot be expressed as the union of two fuzzy separated sets.

Now, we can generalize the definition of fuzzy connected to define fuzzy *sp*-connected as follows.

*Definition 4.10. *A fuzzy set in a is said to be fuzzy *sp*-connected if and only if cannot be expressed as the union of two fuzzy *sp*-separated sets.

Theorem 4.11. *Let be a fuzzy sp -continuous surjective mapping. If is a fuzzy sp -connected subset in then, is fuzzy connected in .*

*Proof. *Suppose that is not connected in . Then, there exist fuzzy separated subsets and in such that .

Since is fuzzy *sp*-continuous surjective mapping, and are fuzzy *sp*-open set in and .

It is clear that and are fuzzy *sp*-separated in . Therefore, is not fuzzy *sp*-connected in , which is a contradiction!!

Hence, is fuzzy connected.

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